Abstract

We describe a nonlinear distortion-tolerant filter for pattern recognition that is optimum in terms of tolerance to input noise and discrimination capability. This filter was derived by minimization of the output energy that is due to the overlapping additive noise and the input scene, and the output of the filter meets the design constraints obtained from the training data set. The performance of this filter was tested with an input scene containing one of the training data sets, a nontraining true target, and a false object in the presence of overlapping additive noise and nonoverlapping background noise. We carried out Monte Carlo runs to measure the statistical performance of the filter and obtained receiver operating characteristics curves to show the detection capabilities of the filter.

© 2002 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  2. A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).
  3. A. McAulay, Optical Computer Architectures: The Application of Optical Concepts to Next Generation Computers (Wiley, New York, 1991).
  4. B. Javidi, J. L. Horner, Real-Time Optical Information Processing (Academic, San Diego, 1994).
  5. J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
    [CrossRef]
  6. A. VanderLugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  7. D. Flannery, J. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511–1527 (1989).
    [CrossRef]
  8. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  9. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  10. B. Javidi, Ph. Réfrégier, P. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1160–1162 (1994).
  11. D. Psaltis, Y. Qiao, H. Li, “Optical face recognition system,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE, 1773, 59–62 (1993).
    [CrossRef]
  12. Ph. Réfréigher, V. Laude, B. Javidi, “Nonlinear joint-transform correlation: an optimal solution for adaptive image discrimination and input noise robustness,” Opt. Lett. 19, 405–407 (1994).
  13. D. Casasent, D. Psaltis, “Position, rotation and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef] [PubMed]
  14. A. Mahalanobis, “Review of correlation filters and their application for scene matching,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Vol. CR65 of SPIE Critical Reviews of Optical Science and Technology (SPIE Press, Bellingham, Wash., 1996), pp. 240–260.
  15. B. Javidi, J. Wang, “Optimum distortion-invariant filter for detecting a noisy distorted target in nonoverlapping background noise,” J. Opt. Soc. Am. A 12, 2604–2614 (1995).
    [CrossRef]
  16. B. Javidi, D. Painchaud, “Distortion-invariant pattern recognition with Fourier-plane nonlinear filters,” Appl. Opt. 35, 318–331 (1996).
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  17. C. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
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  18. J. Caulfield, W. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
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  19. Y. Hsu, H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
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  20. Ph. Réfrégier, F. Goudail, “Decision theoretical approach to nonlinear joint transform correlation,” J. Opt. Soc. Am. A 15, 61–67 (1998).
    [CrossRef]

1998 (1)

1996 (1)

1995 (1)

1994 (2)

B. Javidi, Ph. Réfrégier, P. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1160–1162 (1994).

Ph. Réfréigher, V. Laude, B. Javidi, “Nonlinear joint-transform correlation: an optimal solution for adaptive image discrimination and input noise robustness,” Opt. Lett. 19, 405–407 (1994).

1989 (2)

D. Flannery, J. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef] [PubMed]

1984 (1)

1982 (1)

1980 (1)

1976 (1)

1969 (1)

1964 (1)

A. VanderLugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

1960 (1)

J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

Arsenault, H.

Casasent, D.

Caulfield, J.

Flannery, D.

D. Flannery, J. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Gianino, P. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Goudail, F.

Hester, C.

Horner, J.

D. Flannery, J. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Horner, J. L.

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

B. Javidi, J. L. Horner, Real-Time Optical Information Processing (Academic, San Diego, 1994).

Hsu, Y.

Javidi, B.

Laude, V.

Li, H.

D. Psaltis, Y. Qiao, H. Li, “Optical face recognition system,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE, 1773, 59–62 (1993).
[CrossRef]

Mahalanobis, A.

A. Mahalanobis, “Review of correlation filters and their application for scene matching,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Vol. CR65 of SPIE Critical Reviews of Optical Science and Technology (SPIE Press, Bellingham, Wash., 1996), pp. 240–260.

Maloney, W.

McAulay, A.

A. McAulay, Optical Computer Architectures: The Application of Optical Concepts to Next Generation Computers (Wiley, New York, 1991).

Painchaud, D.

Psaltis, D.

D. Casasent, D. Psaltis, “Position, rotation and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[CrossRef] [PubMed]

D. Psaltis, Y. Qiao, H. Li, “Optical face recognition system,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE, 1773, 59–62 (1993).
[CrossRef]

Qiao, Y.

D. Psaltis, Y. Qiao, H. Li, “Optical face recognition system,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE, 1773, 59–62 (1993).
[CrossRef]

Réfrégier, Ph.

Ph. Réfrégier, F. Goudail, “Decision theoretical approach to nonlinear joint transform correlation,” J. Opt. Soc. Am. A 15, 61–67 (1998).
[CrossRef]

B. Javidi, Ph. Réfrégier, P. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1160–1162 (1994).

Réfréigher, Ph.

Turin, J. L.

J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

VanderLugt, A.

A. VanderLugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).

Wang, J.

Willett, P.

B. Javidi, Ph. Réfrégier, P. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1160–1162 (1994).

Appl. Opt. (7)

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

IRE Trans. Inf. Theory (1)

J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Ph. Réfréigher, V. Laude, B. Javidi, “Nonlinear joint-transform correlation: an optimal solution for adaptive image discrimination and input noise robustness,” Opt. Lett. 19, 405–407 (1994).

B. Javidi, Ph. Réfrégier, P. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1160–1162 (1994).

Proc. IEEE (1)

D. Flannery, J. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Other (6)

D. Psaltis, Y. Qiao, H. Li, “Optical face recognition system,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, J. A. Neff, S. T. Kowel, eds., Proc. SPIE, 1773, 59–62 (1993).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).

A. McAulay, Optical Computer Architectures: The Application of Optical Concepts to Next Generation Computers (Wiley, New York, 1991).

B. Javidi, J. L. Horner, Real-Time Optical Information Processing (Academic, San Diego, 1994).

A. Mahalanobis, “Review of correlation filters and their application for scene matching,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Vol. CR65 of SPIE Critical Reviews of Optical Science and Technology (SPIE Press, Bellingham, Wash., 1996), pp. 240–260.

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Figures (5)

Fig. 1
Fig. 1

(a). Targets used in the simulation. The target from the training data set, the 50°-rotated target from the nontraining data set, and the false object are 15 × 25 pixels in size, and they are normalized so that they have a unity maximum value. (b) Input scene with two targets and one false object as shown in (a). These are buried in zero-mean additive white Gaussian noise with σ = 0.7. (c) Correlation output intensity of the optimum nonlinear distortion-tolerant composite filter for the input scene in (b). The filter has been constructed with the first set of reference targets.

Fig. 2
Fig. 2

(a). Input scene with two targets and one false object as appeared in Fig. 1(a). This scene has nonoverlapping colored background noise that is zero mean and has a standard deviation of 0.1 and a bandwidth of 10. These are buried in zero-mean additive white Gaussian noise with σ = 0.7. (b) Correlation output intensity of the optimum nonlinear distortion-tolerant composite filter for the input scene in (a). The filter has been constructed with the first set of reference targets.

Fig. 3
Fig. 3

Correlation output intensity of the optimum nonlinear distortion-tolerant composite filter with two targets and one false object as in Fig. 1(a) with the same additive noise. However, the nontraining target in the top center has been rotated by 5°, and the filter has been constructed with the second set of reference targets.

Fig. 4
Fig. 4

POE curves of the correlation output. The plot with circles is the POE of the 0°-rotated training target for the filter that uses the second set of reference targets. The plot with squares is the POE of the 5°-rotated nontraining target for the same filter that uses the second set of reference targets. Plots marked with triangles and x symbols are the POE curves of the 0°-rotated training target and the 10°-rotated nontraining target, respectively, for the filter that uses the first set of reference targets. The range of the standard deviations of the additive white Gaussian noise is [0.1, 0.7] with a step size of 0.1.

Fig. 5
Fig. 5

ROC curves of the correlation output. The standard deviations of the additive white Gaussian noise range from 0.6 to 1.2, increasing by 0.1. Only the upper left corners of the curves are depicted to show the details. (a) ROC curves of the 0°-rotated training target for the filter that uses the first set of reference targets. (b) ROC curves of the 10°-rotated nontraining target with the same filter as in (a). (c) ROC curves of the 0°-rotated training target with a filter that uses the second set of reference targets. (d) ROC curves of the 5°-rotated nontraining target with the same filter as in (c).

Equations (29)

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st=i=1Tνirit-τi+nt,
ot=τ=0M-1 ht+τ*sτ,
oi0=t=0M-1 ht*rit=Ci.
k=0M-1 Hk*Rik=MCi,
E1Mk=0M-1 |Hk|2|Nk|2=1Mk=0M-1 |Hk|2E|Nk|2,
1Mk=0M-1 |Hk|2|Sk|2,
wnMk=0M-1 |Hk|2E|Nk|2+wdMk=0M-1 |Hk|2|Sk|2,
k=0M-1 Hk*Rik=k=0M-1ak-jbkcik+jdik =k=0M-1akcik+bkdik+jakdik-bkcik =MCi.
k=0M-1akcik+bkdik=MCi for i=1, 2,, T,
k=0M-1akdik-bkcik=0 for i=1, 2,, T.
wnMk=0M-1 |Hk|2E|Nk|2+wdMk=0M-1 |Hk|2|Sk|2=k=0M-1ak2+bk2Dk
Jk=0M-1ak2+bk2Dk+i=1T λ1iMCi-k=0M-1 akcik-k=0M-1 bkdik+i=1T λ2i0-k=0M-1 akdik+k=0M-1 bkcik.
Jak=2akDk-i=1T λ1icik-i=1T λ2idik=0,
Jbk=2bkDk-i=1T λ1idik+i=1T λ2icik=0.
δkl=1,k=l0,kl.
ak=i=1Tλ1icik+λ2idik2Dk,
bk=i=1Tλ1idik-λ2icik2Dk,
k=0M-112Dki=1Tλ1icikcpk+dikdpk+λ2idikcpk-cikdpk=MCp for p=1, 2,, T,
k=0M-112Dki=1Tλ1icikdpk-dikcpk+λ2idikdpk+cikcpk=0 for p=1, 2,, T.
λ1λ11 λ12  λ1Tt, λ2λ21 λ22  λ2Tt, CC1 C2  CTt, Ax,yk=0M-1ReRxkReRyk+ImRxkImRyk2Dk=k=0M-1cxkcyk+dxkdyk2Dk, Bx,yk=0M-1ImRxkReRyk-ReRxkImRyk2Dk=k=0M-1dxkcyk-cxkdyk2Dk,
λ1tA+λ2tB=MCt,
-λ1tB+λ2tA=0t.
λ1t=MCtA+BA-1B-1,
λ2t=MCtA+BA-1B-1BA-1.
ak+jbk=12Dki=1Tλ1icik+jdik+λ2idik-jcik=12Dki=1Tλ1i-jλ2icik+jdik.
Hk=i=1Tλ1i-jλ2iRik2E|Nk|2+|Sk|2.
Hk=λ1R1k2E|Nk|2+|Sk|2.
Hk=RkE|Nk|2+|Sk|2.
POE=|Ey0|2E1Mt=0M-1 yt2,

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